TG-Hyperbolicity of Virtual Links

A volumish theorem for alternating virtual links

“…The same statement is conjectured for all virtual links, but remains open [4,Conjecture 5.1]. For links as in Theorem 1.1, the hyperbolicity of F × I − K implies that F is the minimal genus representative of K by [1,Theorem 1.2]. The main result of [4] then implies that the reduced alternating surface link diagram has crossing number and writhe that are invariants of the virtual link.…”

“…There is an underlying similarity between the proofs of the two volumish theorems. For alternating links in S 3 , to prove that the twist number is expressed by the sub-extremal coefficients of the Jones polynomial, Dasbach and Lin relied on two key facts: (1) the Jones polynomial of an alternating link is a specialization of the two-variable Tutte polynomial of its Tait graph, and (2) certain coefficients of the Tutte polynomial express the cycle rank of the reduced Tait graph. For alternating links in thickened surfaces, we rely on two similar facts: (1) the reduced Jones-Krushkal polynomial is a specialization of the Krushkal polynomial, which extends the Tutte polynomial to a 4-variable polynomial invariant of graphs on surfaces, and (2) certain coefficients of the Krushkal polynomial express the cycle rank of the reduced Tait graph on the surface (see Definition 3.1).…”